Improve Prim’s algorithm runtime

Assume we run Prim’s algorithm when we know all the weights are integers in the range {1, …W} for W, which is logarithmic in |V|. Can you improve Prim’s running time?

When saying ‘Improving’, it means to at-least: $ $ O(|E|)$ $

My question is – without using priority queue, is it even possible? Currently, we learned that Prim’s runtime is $ $ O(|E|log|E|)$ $

And I proved I can get to O(|E|) when weights are from {1,….,W) when W is constant, but when W is logarithmic in |V|, I can’t manage to disprove/prove it.