This conjecture comes from an observation on simulations of the matrix variate noncentral Beta distribution (similar to [this observation](https://mathoverflow.net/questions/287794/noncentral-matrix-beta-distributions-of-type-i-and-ii), but I open a new question because yet I'm not sure it is exactly the same).


Let $p \geq 1$ be an integer, $a,b > \frac{p-1}{2}$, $\Theta$ a positive scalar $p \times p$-matrix $\Theta = \text{diag}(\theta, \ldots, \theta)$ and $U$ a symmetric $p \times p$-matrix satisfying $0 < U < I\_p$. The conjecture is:
$$
\int\_{S >0} {\det(S)}^{a+b-\frac12(p+1)} 
\exp\left(-\mathrm{tr}\left(\frac{S}{2}\right)\right)
{}\_0\!F\_1\left(b, \frac{1}{2}\Theta S^\frac12 U S^\frac12\right)\textrm{d}S \\
= 2^{a+b}\Gamma\_p(a+b){}\_1\!F\_1(a+b, b, \Theta U).
$$
According to [this paper by Constantine](https://projecteuclid.org/euclid.aoms/1177703863) (page 1280), the integral in the LHS is difficult to evaluate *unless $\Theta$ is a scalar matrix*. This suggests that this integral can be simplified but I don't find the result in the literature. 


Do you have a reference for this result, or a proof?



integral involving hypergeometric function of matrix argument
The zonal polynomial $C\_\kappa$ satisfies $C\_\kappa(SR) = C\_\kappa(S^\frac12 R S^\frac12)$, for $S$ symmetric positive definite and $R$ symmetric, according to (1.5.3) in Gupta & Nagar's book. This implies that the same relation holds for the hypergeometric functions, in view of their definition in terms of the zonal polynomials. Thus, since $\Theta$ is scalar,
$$
{}\_0\!F\_1\left(b, \frac{1}{2}\Theta S^\frac12 U S^\frac12\right) 
= {}\_0\!F\_1\left(b, \frac{1}{2}\Theta S U\right).
$$
Now, it is known that
$$
\int\_{S>0}\exp\bigl(\text{tr}(S)\bigr)\det(S)^{\alpha-\frac12(p+1)}{}\_0\!F\_1(b,ST)\text{d}S \\
= \Gamma\_p(\alpha){}\_1\!F\_1(\alpha,b,T)
$$
(particular case of the known relation between ${}\_m\!F\_n$ and ${}\_{m+1}\!F\_n$, theorem 1.6.2 in Gupta & Nagar's book).


The result follows from these two relations, after the change of variables $S'=S/2$ in the integral of the OP.


