Hello there!

For the MLR estimates to be unbiased (well, to some extent, of course ), two assumptions must be in place -- (a) lack of multicollinearity, and (b) independence of irrelevant alternatives (IIA) (Starkweather, J., & Moske, A. K. (2011). Multinomial logistic regression). The first one is easy to test. The second one could be tested with -mlogtest- in Stata, guess there are similar ones for other software. This was the approach I used in a paper I recently published in a peer-reviewed journal. I wouldn't bother with linearity.