You could ask if it for every odd prime $p$ exist a prime $q<p$ such that $p-q$ is $3$-smooth. My investigstions suggests that an exception should be very big, at least greater than $2^{64}$. Define a function $f(p)$ to be the number of primes $q<p$ such that $p-q$ is $3$-smooth (blue dots below).
It seems to exist lower and upper limits of $f$ of the form $A\cdot\ln p<f(p)<B\cdot\ln p$ as above, where $A\approx 0.5$ and $B\approx 3$.
Below a table of records for the assumed constants $A,B$. The records occure approximately with doubled intervalls and with capricious leaps. It's possible that the leaps doesn't approach zero (fast enough) and that $A\to 0$ and $B\to\infty$ as $p\to\infty$. $$ \begin{array}{r|cc} p & \text{A} & \text{B} \\ \hline 3 & 0.910239226626837 & 0.910239226626837 \\ 5 & 0.910239226626837 & 1.24266986911922 \\ 11 & 0.910239226626837 & 1.66812956569699 \\ 19 & 0.910239226626837 & 1.69811635947554 \\ 29 & 0.910239226626837 & 1.78184522624022 \\ 79 & 0.910239226626837 & 1.83089494831324 \\ 83 & 0.910239226626837 & 2.03673334225281 \\ 181 & 0.910239226626837 & 2.11599620695013 \\ 199 & 0.910239226626837 & 2.2670147284829 \\ 677 & 0.910239226626837 & 2.30143549312477 \\ 1607 & 0.910239226626837 & 2.30286014672168 \\ 1657 & 0.809414677965558 & 2.30286014672168 \\ 2719 & 0.809414677965558 & 2.40262434014324 \\ 3761 & 0.728824004123081 & 2.40262434014324 \\ 3931 & 0.728824004123081 & 2.53725869996085 \\ 12211 & 0.728824004123081 & 2.55045315359463 \\ 14753 & 0.728824004123081 & 2.60438322902193 \\ 30671 & 0.677567576348398 & 2.60438322902193 \\ 43067 & 0.677567576348398 & 2.71777016192333 \\ 54287 & 0.642081661528661 & 2.71777016192333 \\ 60607 & 0.63566061439303 & 2.71777016192333 \\ 88001 & 0.614838505984447 & 2.71777016192333 \\ 148549 & 0.614838505984447 & 2.77109027177284 \\ 272353 & 0.614838505984447 & 2.79667658660832 \\ 348563 & 0.614838505984447 & 2.82096857684845 \\ 740681 & 0.591920639360573 & 2.82096857684845 \\ 769999 & 0.591920639360573 & 2.87734869685204 \\ 1163719 & 0.591920639360573 & 2.93546316868478 \\ 6240473 & 0.539593563616334 & 3.13167747560669 \\ 20818597 & 0.534081609939492 & 3.13167747560669 \\ 42125693 & 0.534081609939492 & 3.13280203029418 \\ *70752683 & 0.534081609939492 & 3.1535791352481 \end{array} $$
Above diagram for records of $A$ (blue) and $B$ (red) as function of the primes where the records occured.
My question is, can someone find records $A<0.5$ or $B>\pi$?
A record greater than $\pi$:
$p=70752683\quad B=3.1535791352481$