The prime-square conjecture for the unit-fraction Maker–Breaker game

Let $k$ be a positive integer and let $p$ be an odd prime such that $k+1=p^2$. Let $f(k,-1)$ denote the least board size for which Maker can force a solution to the unit-fraction equation

$$\frac{1}{x_1}+\cdots+\frac{1}{x_k}=\frac{1}{y}$$

with the variables not necessarily distinct. Prime-square conjecture. If $k+1=p^2$ for some odd prime $p$, then

$$f(k,-1)=k+2.$$

The equality is proved when $k+1$ is an odd prime, but the authors state that they cannot verify the prime-square case and believe it should hold.