Equidistribution conjecture for roots of partition numbers

For each positive integer $n$, let $p(n)$ denote the partition function, and let $\{r\}$ denote the fractional part of a real number $r$. The equidistribution conjecture. For each fixed integer $k\geq 2$, the sequence

$$\left\{\left\{\sqrt[k]{p(n)}\right\}\right\}_{n\geq 1}$$

is equidistributed in $[0,1)$. The source presents strong supporting computational evidence, but no proof is given.