Looking at the Riemann method for the problem, for example here (and Garabedian's PDE book, which I found I had purchased), I think the orig problem is solvable quasi-analytically, in principle, as follows.
1. Take the problem to be:
(*) [$]u_t = \rho \, u_{xy}[$] with
bc: u is given on two adjacent edges and
ic: [$]u(t=0,x,y)=u_0(x,y)[$] is given (arbitrary, as long as compatible with the bc).
2. Take the Laplace transform with respect to time of (*), with [$]U(x,y;s) = \int_0^{\infty} e^{-s t} u(t,x,y) \, dt[$]. This yields
(**) [$]U_{xy} + c \, U = f(x,y)[$], where [$]c = s/\rho[$] and [$]f = -u_0/\rho[$], and where the function [$]U(x,y)[$] is known on two adjacent edges.
3. So, that's a Goursat problem again, and the solution to (**) is given at the link or various books in terms of the edge and initial functions, using the Riemann function [$]R = J_0(\sqrt{4 c (x-\xi)(y-\eta)})[$].
4. Do a numerical Laplace inversion w.r.t. [$]s[$] of the solution in Step 3 to get numbers.
Or don't, as the argument shows what bc and ic's `work', which was actually the original question.