This mode critical because is de-statbilize the thermal boundary layer increasing the heat transfer

Whether equivalent longitudinal mode do not destabilize the thermal boundary layer.

Following

Following

I think Mr. Pempie's reply is the key. I try to expand more based on what I know. The longitudinal mode is composed of a wave that travels in the same direction of the mean flow in a positive and negative way. For this reason, part of its acoustic energy gets expelled from the main combustion chamber from both the nozzle and the injectors. The latter however can come back after interacting with the other engine parts (feed lines, tanks, turbines) causing the feedback loop closing with the combustion itself. The main point here is that due to the nature of the perturbation, the axial derivative of the velocity: \partial u / \partial x is the one that fluctuates the most. This component is not responsible (or very marginally responsible) for the boundary layer at the walls to be thickened, except at the face plate which, however, is not as thermally overloaded as one might expect due to the fact that low temperature propellants are injected in its vicinity and therefore they have a sort of protective effect from the heat loads. On the other hand transverse modes evolve in a direction that is orthogonal to the main flow and for this reason they excite velocity components that are related to \partial u / \partial y and z, hence directly related to the boundary layer thickness at the walls. These walls now are predominantly wet by combustion products that are hot. Every time that the wall experiences a pressure anti-node (highest value), the boundary layer gets squeezed and the thermal loads increase because the temperature gradients at the wall becomes very large locally (\partial T / \partial y and z). Same holds for the mechanical loads. The injector impedance help to dissipate part of the acoustic energy at each cycle however this portion is larger for a longitudinal mode because it evolves in the same direction of the mean flow. For a transverse mode, some portion of the acoustic energy can go in the injectors too, but in general is less because the wave is orthogonal to the mean flow. As a result, a pressure anti-node for a transverse mode can result in a significantly larger mechanical load than a corresponding longitudinal mode. Putting together the thermal and the mechanical loads, you can see now why the transverse mode is much more dangerous than a longitudinal mode. Hope that this helps.

Whether we can say that, the acoustic energy in 1L mode decreases due to loss through boundaries and viscous damping, whereas 1T (tangential modes) mode holds or build on with acoustic energy. The explanation on boundary layer thinning at the point of pressure anti-node leading to higher thermal loads is reasonable and pertinent.