The asymptotic conjecture for the partitioning function $f(A,1)$

Let $f(A,B)$ be the partitioning-function quantity discussed in the paper, with $A$ and $B$ positive integers. The asymptotic conjecture for $f(A,1)$.

$$f(A,1)=A+o(A).$$

The source notes that only $f(1,1)=4$ is known exactly and that the broader bound $f(A,B)\leq A+B+2$ was asked as an open question; the stated asymptotic claim therefore remains open.