Author's conjecture on improving a modified Bessel-function integral inequality

Let $n<1$, let $u\geq 1-n+\sqrt{2(1-n)}$, and define

$$\alpha_{\nu,n}:=\nu+n-\sqrt{(\nu+n)^2-2\nu-1}.$$

For the modified Bessel function of the second kind $K_\mu$, the author conjectures that, for every $x>0$,

$$\int_x^\infty\frac{K_{\nu+n}(t)}{t^\nu}\\,\mathrm{d}t>\frac{K_{\nu+n-\alpha_{\nu,n}}(x)}{x^\nu}.$$

The exponent shift is claimed to be best possible up to the symmetry $K_{-\mu}(x)=K_\mu(x)$: defining

$$\alpha'_{\nu,n}:=\nu+n+\sqrt{(\nu+n)^2-2\nu-1},$$

any $\beta\notin\\{\alpha_{\nu,n},\alpha'_{\nu,n}\\}$ either gives a strictly smaller right-hand side for every $x>0$, or fails to provide a lower bound at some $y>0$. The source also records equality in the corresponding inequality when $n=1$, for all $\nu\in\mathbb{R}$; the conjectured statement itself is presented under $n<1$.