Conjecture on the index of Chebyshev radical extensions

Let $\theta$ be a root of $\theta$'s defining polynomial

and let $ u_p$ denote the $p$-adic valuation. For an odd \prime $p eq\ell$ satisfying $t gleich\pm2\pmod{p^2}$, define the $p$-index by

. Index conjecture. If $p eq\ell$ is an odd \prime for which $t\equiv\pm2\pmod{p^2}$, then

$$\nu_p(\operatorname{ind}(\Phi)):=\operatorname{ind}_p(\Phi)=\left\lfloor\frac{\nu_p(t^2-4)}{2}\right\rfloor\frac{\ell^n-1}{2}.$$

This conjectural formula is intended to determine the contributions of odd primes $p\neq\ell$ to the index of the power basis in the Chebyshev radical extension. The surrounding discussion indicates that the formula follows from the Newton-polygon conjecture below, and its validity is not established in the supplied text.