high & low pass correlated dither noise question
To achieve high-pass shaping , the “DC term” must be 0. Note that in the 2-dice high-pass case, every roll appears twice, once with a positive sign and once with a negative sign, which enforces the DC-free condition.  Any scheme involving more than 2 dice would still need to meet this condition.
The triangular PDF condition is satisfied by the fact that each sum or difference combines 2 rectangular PDF’s. I don’t know how you can get a triangular PDF by combining 3 rectangular PDF’s, regardless of how they are weighted.  

Bob Adams