Suppose I have the following one dimensional moving boundary problem:

$ \frac{\partial h}{\partial t} = \frac{\partial}{\partial x}\left(h^3 \frac{\partial h}{\partial x}\right) \\ h(0,t) = 1\\ h(a(t),t) = 0 \\ \frac{\partial a}{\partial t} = - h^2\frac{\partial h}{\partial x}\left( a(t),t\right) \\ a(0) = 1 \\ h(x,0) = -(-1)^{3/4} \sqrt[4]{x-1}$

I wish to be able to solve this problem using NDSolve to be able to plot a time series of interface spreading along x. Here is my attempt, which fails due to a "The arguments should be ordered consistently." error.

NDSolve[{D[h[x, t], t] == D[h[x, t]^3 D[h[x, t], x], x], 
  h[0, t] == 1, h[a[t], t] == 0, 
  a'[t] == -h[a[t], t]^2 (D[h[x, t], x] /. x -> a[t]), a[0] == 1, 
  h[x, 0] == -(-1)^(3/4) (-1 + x)^(1/4)}, {h, a}, {x, 0, a[t]}, {t, 0,
   10}]

Can this type of moving boundary problem be solved by Mathematica?