  The present paper is a sequel to and generalization of Fung and Seneta (2016)
whose main result gives the asymptotic behaviour as $ u \to 0^{+}$ of
$\lambda_L(u) = P(X_1 \leq F_1^{-1}(u) | X_2 \leq F_2^{-1}(u)),$ when $\bf{X}
\sim SN_2(\boldsymbol{\alpha}, R)$ with $\alpha_1 = \alpha_2 = \alpha,$ that
is: for the bivariate skew normal distribution in the equi-skew case, where $R$
is the correlation matrix, with off-diagonal entries $\rho,$ and $F_i(x),
i=1,2$ are the marginal cdf's of $\textbf{X}$. A paper of Beranger et al.
(2017) enunciates an upper-tail version which does not contain the constraint
$\alpha_1=\alpha_2= \alpha$ but requires the constraint $0 <\rho <1$ in
particular. The proof, in their Appendix A.3, is very condensed. When
translated to the lower tail setting of Fung and Seneta (2016), we find that
when $\alpha_1=\alpha_2= \alpha$ the exponents of $u$ in the regularly varying
function asymptotic expressions do agree, but the slowly varying components,
always of asymptotic form $const (-\log u)^{\tau}$, are not asymptotically
equivalent. Our general approach encompasses the case $ -1 <\rho < 0$, and
covers all possibilities.
