I thought eggs were a good PT as well, but it turns out the molecular weight is too large to penetrate the hair shaft, like Firefox said. Hydrolysed or hydrolyzed protein are what is needed (like Firefox said as well). I think aphogee 2 step PT is said to be good since it contains the hydrolyzed protein. I second the gelatin treatment as well.
High Porosity, Fine, Thin Density, Low Elasticity
Co-wash, leave in, and gel: Alba botanica coconut cond., ogx mousse, &/or super wet look gel.