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

Question

I have a signal a = [a(t₀),a(t₁),a(t₂),..,a(t_N-2),a(t_N-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?