The question should be clearer. At first it wasn't even obvious what method you were referring to, hence Alex clarified that. But there are many variations of this method too. Then for example, what is $\rho$ ? Just because you may know what it refers to, it doesn't mean others do immediately. Because looking at the HW paper Alex provided for example, there's no mention of $\rho$. Looking further I guess I found what you mean (the ratio of the average spacing to the asset spacing at each time-step), but you should've stated it.
Then to the question itself, the 2 or 3 papers I've now seen on this all state that no, they are not the same, with each progressing time step you have to use more discretized average states/values. Which is logical because as the tree's asset range expands, so should the possible average range. You should just make sure that at every time step this range covers the maximum and minimum possible average, based on the asset values the tree has taken thus far (plus the running average in your case).
Funnily enough the first result that comes up searching for forward shooting grid method is a matlab code that implements it for arithmetic Asians, albeit with no running average. This page though seems to have been taken down recently, maybe your supervisor owns it? :) Of course there is still the cached version...
Finally, it may help you indirectly if I told you the right price for this option is 0.11274 (calculated with an equivalent PDE method). As you increase the number of time periods in the lattice you should be converging to that, otherwise something's wrong.