Does universal property of simply connected covering spaces require local path connectedness?

Question

I am trying to find out if the property that simply connected covering spaces cover other connected covers (the universal property) depends on local path connectedness. I know that the universal property of simply connected covers can be proven from the lifting criterion, but all the statements of the lifting criterion I have seen depend on local path connectedness. According to Wikipedia (https://en.wikipedia.org/wiki/Covering_space#Universal_covers), the universal property does not depend on local path connectedness, but I would like to double check this. I am trying to avoid getting too deep into the weeds and will accept essentially a yes/no answer, but would also appreciate a vague idea where a proof of this fact comes from, assuming that it is true.

Answer 1

I think the following two questions and its answers contain relevant information:

1. The universal cover covers any connected cover
   Here you will learn that some authors define the concept of a universal covering by the property that it covers any covering $p' : E' \to X$ with a connected $E'$ which is a nice explanation of the name. It is then a theorem that a simply connected covering of a connected and locally path connected X is a universal covering.

2. Classification of covering spaces for spaces that are not locally path connected: counterexamples?
   Here you will learn that the Warsaw circle $X$ (which is simply connected, but not locally connected) has $id : X \to X$ as a simply connected covering space. But it is not a universal covering space in the above alternative sense.