Consecutive-value conjecture for supersingular elliptic curves

For each $n\in\mathbb{Z}_{>0}$, there should exist a prime $p$ such that, for every integer $m$ with $1\leq m\leq n$, there is a supersingular elliptic curve $E$ defined over $\overline{\mathbb{F}}_p$ with $\delta_E=m$. Consecutive-value conjecture. For each $n\in\mathbb{Z}_{>0}$, there exists a prime $p$ such that for all $1\leq m\leq n$, there is a supersingular elliptic curve $E$ defined over $\overline{\mathbb{F}}_p$ with $\delta_E=m$. The conjecture is verified in the paper for every $n\leq48$ using $p=234959$, but no general proof is given.