How to initialize shift registers in polyphase filter based resamplers to avoid or compensate for group delay?

I have a signal a = [a(t0),a(t1),a(t2),..,a(tN-2),a(tN-1)]

I'm using a polyphase filter based resampler with filter length $L$ of $8$, fixed interpolation rate $I$ of $10000$ and decimation rate of $D$ to resample the signal $a$. After resampling, I'm seeing a group delay of $N = \frac{L \times I}{2 D}$ samples.

I'm able to partially compensate for the group delay by initializing shift register S with the samples of the signal a in the following configuration,

S = [a(t0),a(t1),a(t2),a(t3),a(t0),a(t1),a(t2),a(t3)]

The above configuration compensates for the group delay, but does not reconstruct the signal from t0 to t4 accurately. What would be the ideal way to initialize the shift register to reconstruct the signal with better accuracy?