Analysis of this general endgame appears in Staunton's Handbook, in the section "King and Three Passed Pawns against King and Three Passed Pawns."

In the preceding section of the book, "King against Three Passed Pawns," Staunton provides analysis indicating that a lone king battling the three connected passers in isolation (without assistance from their own king) can stop a promotion by getting to the so-called master square, which are here g3 for White, b6 for Black.

When considering these KPPP vs. KPPP endgames, he asserts a general rule about the kings' starting points for those positions in which one or both kings are within 3 squares of their respective master squares:

Victory will be in the hands of the party who can first play his King into its master square.

Staunton goes into some detail explaining the general logic behind play, and gives some examples. I won't try to elaborate here, but instead refer the reader to Staunton's text if interested. According to the rule, the OP's position would indeed be a win for White.

Among the examples Staunton considers, one is "Greco's position," which is identical to the OP's but with kings on e1 and e8. According to Staunton, that position was long considered drawn, but analysis given by Szén revealed it to be a win (in accordance with the rule above). Staunton further considers the position with kings on d1 and e8, calling it "Szén's position," and presenting analysis he credits to Evans (consistent with bof's comment to the OP) showing this also to be a win for White (again in accordance with the master square rule).

For those interested in more beyond Staunton: there is a paper by Noam Elkies applying combinatorial game theory to chess endings, and which in particular considers cases where pawn endgames essentially reduce to distinct subgames wherein mutual Zugzwang might arise. It deals with some of these KPPP vs. KPPP endings near the end, including Szén's position and a nice study by Behting:

[fen "1k6/1P6/2P5/P7/5ppp/8/6K1/8 w - - 0 1"]
[White "White"]
[Black "to Win"]

Analysis of this general endgame appears in Staunton's Handbook, in the section "King and Three Passed Pawns against King and Three Passed Pawns."

In the preceding section of the book, "King against Three Passed Pawns," Staunton provides analysis indicating that a lone king battling the three connected passers in isolation (without assistance from their own king) can stop a promotion by getting to the so-called master square, which are here g3 for White, b6 for Black.

When considering these KPPP vs. KPPP endgames, he asserts a general rule about the kings' starting points for those positions in which one or both kings are within 3 squares of their respective master squares:

Victory will be in the hands of the party who can first play his King into its master square.

Staunton goes into some detail explaining the general logic behind play, and gives some examples. I won't try to elaborate here, but instead refer the reader to Staunton's text if interested. According to the rule, the OP's position would indeed be a win for White.

Among the examples Staunton considers, one is "Greco's position," which is identical to the OP's but with kings on e1 and e8. According to Staunton, that position was long considered drawn, but analysis given by Szén revealed it to be a win (in accordance with the rule above). Staunton further considers the position with kings on d1 and e8, calling it "Szén's position," and presenting analysis he credits to Evans (consistent with bof's comment to the OP) showing this also to be a win for White (again in accordance with the master square rule).

For those interested in more beyond Staunton: there is a paper by Noam Elkies applying combinatorial game theory to chess endings, and which in particular considers cases where pawn endgames essentially reduce to distinct subgames wherein mutual Zugzwang might arise. It deals with some of these KPPP vs. KPPP endings near the end, including Szén's position and a nice study by Behting:

[fen "1k6/1P6/2P5/P7/5ppp/8/6K1/8 w - - 0 1"]
[White "White"]
[Black "to Win"]