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We begin by investigating relationships between two forms of Hilbert-Schmidttwo-re[al]bit and two-qubit "separability functions"--those recently advancedby Lovas and Andai (J. Phys. A 50 [2017] 295303), and those earlier presentedby Slater (J. Phys. A 40 [2007] 14279). In the Lovas-Andai framework, theindependent variable $\varepsilon \in [0,1]$ is the ratio $\sigma(V)$ of thesingular values of the $2 \times 2$ matrix $V=D_2^{1/2} D_1^{-1/2}$ formed fromthe two $2 \times 2$ diagonal blocks ($D_1, D_2$) of a $4 \times 4$ densitymatrix $D$. In the Slater setting, the independent variable $\mu$ is thediagonal-entry ratio $\sqrt{\frac{\rho_{11} \rho_{44}}{\rho_{22}\rho_{33}}}$--with, of central importance, $\mu=\varepsilon$ or$\mu=\frac{1}{\varepsilon}$ when both $D_1$ and $D_2$ are themselves diagonal.Lovas and Andai established that their two-rebit "separability function"$\tilde{\chi}_1 (\varepsilon )$ ($\approx \varepsilon$) yields the previouslyconjectured Hilbert-Schmidt separability probability of $\frac{29}{64}$. We areable, in the Slater framework (using cylindrical algebraic decompositions [CAD]to enforce positivity constraints), to reproduce this result. Further, we newlyfind its two-qubit (yielding $\frac{8}{33}$), two-quater[nionic]-bit (yielding$\frac{26}{323}$) and "two-octo[nionic]-bit" (yielding $\frac{44482}{4091349}$)counterparts. Then, we find a Lovas-Andai "master formula",$\tilde{\chi_d}(\varepsilon)= \frac{\varepsilon ^d \Gamma (d+1)^3\,_3\tilde{F}_2\left(-\frac{d}{2},\frac{d}{2},d;\frac{d}{2}+1,\frac{3d}{2}+1;\varepsilon^2\right)}{\Gamma \left(\frac{d}{2}+1\right)^2}$ encompassing both even and oddvalues of $d$. C. Koutschan, then, using his HolonomicFunctions program,develops an order-4 recurrence satisfied by the predictions of the severalformulas, establishing their equivalence.

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Master Lovas-Andai and Equivalent Formulas Verifying the $\frac{8}{33}$  Two-Qubit Hilbert-Schmidt Separability Probability and Companion  Rational-Valued Conjectures