The commutation conjecture for generic commuting matrix equations

Let $K$ be a field, let $g$ be a nonzero polynomial in the noncommuting indeterminates $u_1,\ldots,u_{k+1}$ such that

$$g(u_1,\ldots,u_k,0)=0,$$

for every $i\leq k$, the indeterminate $u_i$ appears exactly once in $g$, and the degree of $g$ with respect to $u_i$ is $1$. Let $A_1,\ldots,A_k,B$ be generic commuting $n\times n$ matrices, and consider

$$g(A_1,\ldots,A_k,X)+B=0_n$$

in the unknown $X\in\mathcal{M}_n(\overline{K})$. The commutation conjecture. Every solution $X$ commutes with each $A_i$ and with $B$. Numerical experiments, including experiments for a quartic equation when $n=3$, suggest this statement, but the supplied text does not establish it in general.