Infinitude of smooth shifted primes in arithmetic progression with fixed moduli

Question

Let $P(n)$ denote the largest prime factor of $n$. I wanted to know if an analogue of Baker-Harman estimate for smooth shifted primes in arithmetic progressions with fixed moduli is there in literature.

Baker-Harman proved that there exist infinitely many primes $p$ such that $P(p-1)<p^{0.2961}$ and the exponent was recently improved to $0.2844$ by Lichtman recently. Has someone considered the problem of obtaining an inifnitude of such primes which are of form $1\pmod{a}$ for some fixed modulus $a$?

Thanks in advance for any help.

Answer 1

The only relevant result I found is a paper by Banks, Harcharras, and Shparlinski. They did not establish existence of such smooth numbers but instead gave an upper bound.

Let

$$ \pi_h(x,y;q,a)=\#\{p\le x:P(p-h)<y\wedge p\equiv a\pmod q\}. $$

Then uniformly for $x\ge y\ge2$ and $q\ge1$, they proved that

$$ \pi_h(x,y;q,a)\ll{u\rho(u)\over\varphi(q)}\prod_{\substack{p|h\\p>2}}{p-1\over p-2}\pi(x). $$