We have $$T(4,1,x)\sim t(x):=e^{-x}/x^3$$ as $x\to\infty$. This can be obtained from formulas (29) and (30) of the linked paper by expressing $T(3,1,x)$ and then $T(4,1,x)$ in terms of $\Gamma(a,x)$ and its first two derivatives in $a$ at $a=1$, and then using repeatedly integration by parts to obtain enough terms of the asymptotics of these two derivatives. I can provide details later.
Here is the graph $\{(x,\frac{T(4,1,x)}{t(x)})\colon20\le x\le200\}$:
