Equality of multiplicity bounds for polynomial and piecewise linear coefficients

Let $\mu_{1}(m,n)$ be the maximum multiplicity of $z=0$ when $B(t)$ and $A(t)$ are polynomial functions of degrees $m$ and $n$, respectively. Let $\mu_{2}(m,n)$ be the maximum multiplicity of $z=0$ when $B(t)$ and $A(t)$ are continuous piecewise linear functions with $m$ and $n$ segments, respectively; for $\mu_{2}$, the segments are connected at $\frac{k}{n}$ for $k=1,2,\ldots,n-1$. Consider equations (1.1) and (1.4) from the paper. Multiplicity equality conjecture. For the equations (1.1) and (1.4),

$$\mu_{1}(m,n)=\mu_{2}(m,n)$$

for all $m$ and $n$. The preceding results verify this equality for several low-degree cases, but its validity for all $m$ and $n$ remains open.