The result you get using most brushes in Illustrator do this, it's the nature of the beast.
If you want to have the stroke to look uniform, copy the rectangle and paste in place (Ctrl+Shift+V)
Then go to Object>Transform>Reflect>Horizontal.
Someone else may well come along with a better way...
Bob.
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as Bob said the brush is designed to do this
as a quick fix I would do the same as he did - on my version of illustrator it is 'paste in front' or paste in back' [not paste in place], it probably does not matter which you use as long as the fill is in the back one
you can group these, but if it is not acceptable to have two objects then I think you need to use a brush that does not fade, or make a custom one yourself
[however as Bob also said there may be an easier way we don't know about, I don't claim to be an expert in illustrator ]
There are a few other ways to do it but you have been shown the quickest. Use the Pen Tool to add 2 points just before it tapers then use direct selection tool to pull 1 of the points over onto corner (messy). As line is vector go into line with Direct Selection too and pull tapered points up and down to broaden the taper (slow).
Design is thinking made visual.
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