Answer: As far as we know, many Japan professional Renju professionals claimed to prove them. In academic society, it is said that Black has been proved to win in the free style by Allis (1994) and Allis, Van den Herik and Huntjens (1996), and under Renju restrictions by Wágner and Virág (2001).

This often caused misunderstanding between Renju players and academic people. The key difference is as follows. Academic people need to prove it clearly for all variations, while Renju professionals (similarly for Go professionals) normally would not elaborate the details. We quote a paragraph (the second paragraph in Section 5.1) from the Thesis by Allis (1994).

In Japan professional renju players (renju being a complicated variant of go-moku) have studied go-moku in detail and have stated that the player to move first (black) has an assured win (Sakata and Ikawa, 1981). These statements are sometimes accompanied by a list of main variations, such as the 32-page analysis in Sakata and Ikawa (1981). Close examination of these analyses reveals that in each position only a small number of white moves are analyzed.  For example, after black's first move at the center of a 15x15 board, white has 35 distinct moves, of which 2 are adjacent to black's first move, ignoring symmetrically equivalent moves. In Sakata and Ikawa (1981) only the variations after 2 moves adjacent to black's first move are discussed. As far as we know, prior to this work no complete proof of black's win in go-moku has been published.

References:

Allis, L. V. (1994). Searching for solutions in games and artificial intelligence, Ph.D. Thesis, University of Limburg, Maastricht.

L.V. Allis, H.J. van den Herik, M.P.H. Huntjens, Go-Moku solved by new search techniques, Comput. Intelligence: An Internat. J. 12 (1) (1995) 7–24.

Sakata, G. and Ikawa, W. (1981). Five-In-A-Row. Renju. The Ishi Press, Inc., Tokyo, Japan.

Wágner, J., Virág, I. (2001) Solving Renju, ICGA Journal, Vol. 24 (1) 30–34.