7. Let $r_{a}, r_{b}, r_{c}$ denote the radii of the exscribed circles corresponding to the sides of lengths $a, b, c$ respectively, and $R, p$ and $S$ denote the circumradius, semiperimeter, and area of the given triangle. It is well-known that $r_{a}(p-a)=r_{b}(p-b)=r_{c}(p-c)=S=\sqrt{p(p-a)(p-b)(p-c)}=\frac{abc}{4R}$. Hence, the desired inequality $r_{a} r_{b} r_{c} \leq \frac{3 \sqrt{3}}{8} abc$ reduces to $p \leq \frac{3 \sqrt{3}}{2} R$, which is by the law of sines equivalent to
$$\sin \alpha + \sin \beta + \sin \gamma \leq \frac{3 \sqrt{3}}{2}$$
This inequality immediately follows from Jensen's inequality, since the sine is concave on $[0, \pi]$. Equality holds if and only if the triangle is equilateral.