The polynomial bound on integer zeros of straight-line computable polynomials

Let a straight-line program for a polynomial in one variable be a list

$$s_0=1,\quad s_1=x,\quad s_2,\ldots,s_{\tau},$$

where each $s_i$, for $i\geq 2$, is one of $s_j+s_k$, $s_j-s_k$, or $s_js_k$ for some $j,k<i$. For $f\in\mathbb Z[x]$, let $\tau(f)$ be the least length of a straight-line program computing $f$, and let $n(f)$ denote the number of distinct integer zeros of $f$. Integer-zero bound conjecture. There is a constant $a>0$ such that, for every univariate polynomial $f\in\mathbb Z[x]$,

$$n(f)<\tau(f)^a.$$

Such a bound would connect the arithmetic complexity of straight-line programs with the number of integer roots and is presented in the context of complexity over $\mathbb C$ and the separation of polynomial-time and nondeterministic polynomial-time computation.