How large will the smallest counter-example probably be?

Question

Here :

Primes $p$ of the form $10^{m}(2^{k}−1)+2^{k-1}−1$, where $m$ is the number of decimal digits of $ 2^{k-1}$.

it is conjectured that the number $$(2^k-1)\cdot 10^m+2^{k-1}-1$$ where $m$ is the number of decimal digits of $2^{k-1}-1$ (See also the title), is never prime when it is of the form $7s+6$. Amazingly, primes with the other residues exists although the residue $6$ occurs twice often than $1$.

Upto $k=131\ 000$, there is no counter-example, but probably a counter-example exists because the growth rate is roughly $4^k$ and $1$ out of $9$ $k's$ lead to a number of the form $7s+6$.

How large will the smallest counterexample probably be ?