An almost strong relation

Let $\mu$ be a strong limit singular cardinal. We prove that if $2^{\mu }>{\mu }^{+}$ then $\left(\genfrac{}{}{0px}{}{{\mu }^{+}}{\mu }\right)\to {\left(\genfrac{}{}{0px}{}{\tau }{\mu }\right)}_{<\mathrm{c}\mathrm{f}(\mu )}$ for every ordinal $\tau <{\mu }^{+}$. We obtain an optimal positive relation under $2^{\mu }={\mu }^{+}$, as after collapsing $2^{\mu }$ to ${\mu }^{+}$ this positive relation is preserved.