I think that matter wave could be both longitudinal as well as transverse!!!
Even if, by this time, any scientist anywhere proved that it is either longitudinal or transverse, may be corrected by somebody else in future with this change - it is both longitudinal as well as transverse.
As we know generally, the concept of matter wave is just the idea of probability of finding a matter at a particular position as the square of amplitude of the wave. Then, this probability is equally valid with a longitudinal or a transverse wave.
Another fact we all know is that, thoughts come first and then it changes into actions and results. In the same way, we could say that in the universe, field and waves could be the first existing ones and then came the matter. A wave is a disturbance produced in a medium and hence, the field itself could be the first medium, with equal probability for a longitudinal and a transverse wave. Both may not be existing simultaneously. However, since the speed is too high, even if we look into the origin of the wave itself, we may not be able to distinguish correctly the time delay of production of these two different types and we may feel that both are simultaneously existing. Matter produced from longitudinal waves (or the probability of finding a matter in the longitudinal waves) and matter produced from a transverse waves (the probability of finding a matter in the transverse wave) could have different properties or characteristics.
These topics need further research and I am sure that just the concepts of physics would not be enough to reach the result - In physics we know about the fields, forces and all. As an example, we know that the like fields from a charge or magnet will repel each other. We do not know two fields, interacting with each other, changes its nature (from attraction to repulsion or the reverse) due to the circumstances or surroundings. However, we all know that, our thoughts and attitudes, which are also nothing but fields, created by the complex brain neurons and fields, will change the nature. Same thing could be a matter of love or attraction for the same person at one time and may be a matter of hatred at another time.
Instead of going ahead through a single branch of science, we should think of joint researches by experts from different fields.