Least denominator of rationals in explicit intervals coming from a Ramanujan series for $1/\pi$

Let

$N_n=\binom{2n}{n}^3(42n+5)$.

Define

$L_m=\sum_{n=0}^m \frac{N_n}{2^{12n+4}}$

and

$U_m=L_m+\frac{4N_{m+1}}{3\cdot 2^{12m+16}}$.

These are rational intervals coming from a Ramanujan series for $1/\pi$. For this question, I only want to study rational points inside these explicit intervals.

Let

$Q_m=\min\{q\ge 1:\text{ there is }p\in\mathbb Z\text{ with }\gcd(p,q)=1\text{ and }p/q\in[L_m,U_m]\}$.

So $Q_m$ is the least denominator of a reduced rational number inside $[L_m,U_m]$.

The interval length is

$U_m-L_m=\frac{4N_{m+1}}{3\cdot 2^{12m+16}}$.

Using the usual estimate for the central binomial coefficient, this length is about

$m^{-1/2}2^{-6m}$.

So the square-root scale suggests that the first rational in the interval should usually have denominator around $2^{3m}$.

I would like to prove the following weaker statement:

For every $\eta>0$, we have

$Q_m>2^{(3-\eta)m}$

for all sufficiently large $m$.

Equivalently, for every fixed $\eta>0$, I want to prove that for all large $m$ there are no coprime integers $p,q$ such that

$q\le 2^{(3-\eta)m}$

and

$L_m\le p/q\le U_m$.

After clearing denominators, this becomes the following integer-strip problem. Put

$D_m=3\cdot 2^{12m+16}$,

$A_m=3\sum_{n=0}^m N_n2^{12(m-n)+12}$,

and

$R_m=4N_{m+1}$.

Then $L_m=A_m/D_m$ and $U_m=(A_m+R_m)/D_m$. Thus $p/q\in[L_m,U_m]$ is equivalent to

$0\le D_mp-A_mq\le R_mq$.

So the question is:

For fixed $\eta>0$, can one prove that for all large $m$ there is no solution in integers $p,q$ with $1\le q\le 2^{(3-\eta)m}$ and

$0\le D_mp-A_mq\le R_mq$?

A weaker version would also be useful. For a set $A\subseteq\mathbb N$, define its upper asymptotic density by

$d^+(A)=\limsup_{N\to\infty}\frac{|A\cap[N]|}{N}$.

Or even weaker: Is it possible to prove that there exist $a>0$ and $s>0$ such that

$d^+({m:Q_m\le 2^{am}})\le 1-s$?

I am looking for methods or references. Any thoughts would be very helpful.