Part of an infinite series, where most tilings in this series are not flc, this one is the exception.
The reason is that the inflation factor is a - real - PV number.
By an argument in [PR] this forces flc.
Interestingly, the shape of the tiles can vary.
That is, there is one free parameter
$0 < l < 1+s$, and the smallest prototile is the triangle with sides
$s$ the largest root of
In particular, one can have obtuse triangles, as well as rectangular ones or acute ones as prototiles.
Priebe-Frank, N and Robinson, E A jr
Generalized beta-expansions, substitution tilings, and local finiteness