This post is related to questions asked  [here](https://mathoverflow.net/q/419970/160943) and [here](https://mathoverflow.net/q/80865/160943). However, I include the relevant background on least prime in arithmetic progressions presented [here](https://mathoverflow.net/q/419970/160943) for benefit of the reader.

By Linnik's theorem there are constants $c,L$ such that for every $d \geq 2$ and $1\leq a<d$ with $(d,a) = 1$ the least prime $p_{\text{min}}(d,a)$ congruent $a$ modulo $d$ satisfies
$$
	p_{\text{min}}(d,a) \leq c d^L.
$$
Currently the best known value for the exponent is $L=5$ (Xyloris).
On the extended Riemann hypothesis or the generalized Riemann hypothesis, we have $L = 2+\epsilon$ for every $\epsilon > 0$. 
A folklore conjecture (sometimes attributed to Chowla, sometimes to Heath-Brown) states that $L = 1+\epsilon$ for all $\epsilon$.

Fix a prime number $p$ and fix $a$, say $a=1$. For my purposes the only relevant case is when $d$ a power of the fixed prime number $p$. In this case stronger results are known.
Let $L(p)$ be defined as
$$
 L(p) = \limsup_{j \to \infty} \frac{\log(p_{\text{min}}(p^j,1))}{j \log(p)}.
$$
In other words $L(p)$ is the infimum over all real numbers $L> 0$ such that $p_{\text{min}}(p^j) \leq c_L p^{jL}$ for some $c_L > 0$ and all $j \geq 1$.
Barban, Linnik and Tshudakov proved $L(p) \leq \frac{8}{3}$. Gallagher established $L(p) < 2.5$ and Huxley improved this to $L(p) \leq 2.4$. The best known bound can be found in a paper of [Banks-Shparlinski](https://link.springer.com/article/10.1007/s11854-019-0060-4): $L(p) < 2.1115$.

I wanted to ask if we could possibly strengthen the upper bound on $L(p)$ to the conjectural bound $1+o(1)$ under Generalised Riemann Hypothesis. One can achieve the bound  $p(a,d)\ll d^{1+o(1)}$  if one assumes the Montgomery's conjecture as answered in this [post](https://mathoverflow.net/a/1498/160943) but as sharper upper bounds have been proven for $L(p)$ than the general case, is there any possible way to get the conjectured upper bound on GRH?
Thanks in advance for any assistance.