On finitely based $\mathrm{T}$-spaces of free Lie nilpotent algebras of rank $2$

It is proved that in free Lie nilpotent n-class algebra $F_2^{(n)}$ of rank $2$ over the field of characteristic $p \ge n\ge 4$ there exists a finite decreasing series of $\rm T$-ideals $T_0 \supseteq T_1\supseteq \dots T_k\supseteq T_{k+1}=0$, such as the $T_0=T^{(3)}$ – $\rm T$-idel, generated by the commutator $[x_1,x_2,x_3]$, and factors $T_i/T_{i+1}$ do not contain the proper $\rm T$-spaces. This implies that every $\rm T$-space of the algebra $F_2^{(n)}$ which contained in the $\rm T$-ideal $T ^ {(3)}$ has a finite system of generators.
This result is an answer to the question of A.V. Grishin, formulated in the work A.V. Grishin, On $\rm T$-spaces in a relatively free two-generated Lie nilpotent associative algebra of index 4, J. Math. Sci. 191:5 (2013), 686–690.