If $L_n(x)$ is the nth Laguerre polynomial, let $A_{rst} (\alpha ) = \int _0^\infty e^{ - \alpha x} L_r (x)L_s (x)L_t (x)dx$. It has recently been shown that $A_{rst} (\alpha ) > 0$ for $\alpha \geqq 2,r,s,t = 0,1, \cdots $, while $( - 1)^{r + s + t} A_{rst} (\alpha ) \geqq 0$ for $0 0$ for $r \geqq t$. The complete conjecture has not yet been proved, but it is established here for the cases $0 \leqq t \leqq 10$, $r \geqq t$, by obtaining, for each such t, an explicit expression for $A_{rrt} ({3 / 2})$ as a function of r. The numbers $A_{rrt} ({3 / 2})$ have been evaluated by means of recurrence relations up to $r = t = 500$ and the conjecture has been found to hold. In addition $A_{rrt} ({3 / 2})$ has been evaluated asymptotically as $r \to \infty $ both for fixed t and for $t = r$. The asymptotic expressions verify the conjecture. The main technique used is that of recurrence relations and generating functions, with the asympt...