Ulas's conjecture on the distribution of roots of m-ary partition polynomials

Let $p_m(n,t)$ denote the $m$-ary partition polynomial, and for natural numbers $m\geq 2$ and $n$ define the set of its nonzero roots

$$Z_{m,n}:=\{z\in\mathbb{C}\mid p_m(n,z)=0\}\setminus\{0\}.$$

For $N\in\mathbb{N}$, set

$$Z_m^{(N)}:=\bigcup_{n>N}Z_{m,n}.$$

Ulas's conjecture. For every $\varepsilon>0$ and every $m\geq 2$, there exists $N$, possibly depending on $\varepsilon$ and $m$, such that

$$Z_m^{(N)}\subseteq\{z\mid 1-\varepsilon<|z|<1+\varepsilon\}.$$

Moreover, for every $m\geq 2$,

$$\#\left(Z_m^{(0)}\cap\{z\mid |z|>1\}\right)=\#\left(Z_m^{(0)}\cap\{z\mid |z|<1\}\right)=\infty.$$

Thus, there are infinitely many roots of $m$-ary partition polynomials inside and outside the unit circle. The first assertion says that, for each fixed $m$, the nonzero roots from sufficiently large indices lie arbitrarily close to the unit circle. The additional assertion predicts infinitely many roots on each side of that circle; the source presents both assertions as a problem/conjectural belief, and does not provide a resolution.