# Distribution of i.i.d. random variables $X$ and $Y$ if $XY \sim \text{Beta}(\alpha, \beta)$

This is a follow-up to the question Square root of a Beta(1,1) random variable, which received two great answers.

If $$XY \sim \text{Beta}(\alpha, \beta)$$, and $$X$$ and $$Y$$ are two independent identically distributed random variables, do we know their distribution? If there is no simple general solution, can we figure out the distribution of $$X$$ and $$Y$$ for the case $$XY \sim \text{Beta}(1, 1)$$?

From my search so far I guess this is a deconvolution problem for which no readily worked-out solution, such as the Kumaraswamy distribution in the previous question, exists. However, I would appreciate to be proven wrong in this regard.

In the case of $$Z=XY\sim \text{Beta}(\alpha,1)$$, the moment generating function (mgf) of $$-\ln(XY)=-\ln X-\ln Y$$ is \begin{align} M_{-\ln(XY)}(t) &= E(e^{-t\ln Z}) \\ &=E(Z^{-t}) \\ &=\int_0^1 z^{-t}\alpha z^{\alpha-1}dz \\ &= \frac{1}{1-t/\alpha} \end{align} which is the mgf of an exponentially distributed random variable with rate parameter $$\alpha$$.
Thus, if $$X$$ and $$Y$$ are iid, this means that $$-\ln X$$ and $$-\ln Y$$ must be Gamma distributed with shape parameter $$1/2$$ and rate parameter $$\alpha$$ (see wikipedia). Backtransforming, the pdf of $$X$$ and $$Y$$ is $$f(x)=\sqrt{-\frac\alpha{\pi\ln x}}x^{\alpha-1}$$ for $$0\le x\le 1$$.
A similar calculation in the general Beta$$(\alpha,\beta)$$-case leads to $$M_{-\ln(XY)}(t)=\frac{\Gamma(\alpha+t)\Gamma(\alpha+\beta)}{\Gamma(\alpha+\beta+t)\Gamma(\alpha)}$$ and the mgf of $$-\ln X$$ and $$-\ln Y$$ would need to be the square root of this. But that is not the mgf of any known distribution and such a distribution (that when convolved with itself produces the target density) may not even exist.