This question has been motivated by weak* completeness of distributions.

According to the answer in the above post, any barrelled locally convex topological vector space $E$ satisfies the uniform boundedness principle for its continuous dual space $E'$.

My question is

Does the uniform boundedness principle hold for jointly continuous multilinear functionals on $E$ as well?

I have looked for any existing result on this, but it seems a bit elusive for me.

Could anyone please provide any information? I will move it to ME if this is not research-level question.