Skalba's density conjecture for primes dividing binary sums

Let $T_k$ denote the set of primes $p$ dividing an integer of the form $2^{a_1}+\dots+2^{a_k}+1$, where $a_1,\ldots,a_k\in\mathbb{N}$, and let $T_k(X)$ be the number of such primes $p\leq X$. Skalba's conjecture.

$$T_2(X)\sim\frac{X}{\log X}.$$

This asserts that almost all primes divide an integer of the form $2^a+2^b+1$. Skalba proved a partial result toward the conjecture, while the conjecture itself remains open.