Jones's polynomial Cunningham-chain reducibility conjecture

Let $k\geq 1$ be an integer, and define

$$t=2\left\lceil\frac{k+1}{2}\right\rceil+1,$$ $$m=2\left(\frac{2^t+1}{3}\right).$$

For an integer $j\geq t-k$, consider the polynomial

$$x^j+x^{j-1}+\cdots+x+m.$$

Jones's polynomial Cunningham-chain reducibility conjecture. The polynomial is reducible over $\mathbb{Q}$ if and only if $j=t$. This conjecture proposes an explicit family of initial polynomials whose associated polynomial Cunningham chains have a prescribed reducibility pattern; the supplied text gives no resolution status.