Existence of Strongly Locally Compact Space which is not Weakly Sigma-Locally Compact

Theorem
There exists at least one example of a strongly locally compact topological space which is not also a weakly $\sigma$-locally compact space.

Proof
Let $T$ be an uncountable discrete space.

From Discrete Space is Strongly Locally Compact, $T$ is a strongly locally compact space.

From Uncountable Discrete Space is not $\sigma$-Compact, $T$ is not a weakly $\sigma$-compact space.

By definition, it follows that $T$ is not a weakly $\sigma$-locally compact space.

Hence the result.