This is a question about Faltings' $p$-adic Eichler-Shimura isomorphism from his 1987 article "Hodge-Tate structures and Modular Forms".

Let $N\ge5$, $k\ge2$ be integers. Denote by $X(N)$ the proper modular curve of full level $N$ (over $\mathbb Q$, say), $f\colon\overline E(N)\rightarrow X(N)$ the universal generalized elliptic curve over it and by $e\colon X(N)\rightarrow E(N)$ the unit section. Further let $\omega=e^*\Omega^1_{E(N)/X(N)}$ and $\Omega^1=\Omega^1_{X(N)}$. Let $Y(N)$ be the open modular curve and denote the universal elliptic curve over it still by $f$. Put $V=H^1_{\mathrm{p}}(Y(N)\times\overline{\mathbb Q},\operatorname{Sym}^{k-2}R^1f_*\mathbb Z_p)$, $W_0 = H^0(X(N),\omega^{k-2}\otimes\Omega^1)$ and $W_{k-1}=H^1(X(N),\omega^{2-k})$.

Faltings' $p$-adic Eichler-Shimura isomorphism, which is Thm. 6 (iii) in his paper, states that canonically $$ \mathbb C_p\otimes V(1)\cong \mathbb C_p\otimes W_0 \oplus \mathbb C_p(k-1)\otimes W_{k-1}. $$

It seems to be well-known that this Eichler-Shimura isomorphism "is" the $p$-adic comparison isomorphism for the modular motive ${}^N_kW$ introduced by Scholl in his 1990 paper "Motives for Modular Forms". In trying to make this statement precise, I obtained the following.

The motive introduced by Scholl which he calls ${}^N_kW$ has $V$ as its $p$-adic étale realization and $W:=W_0\oplus W_{k-1}$ as its Hodge realization, with the $W_i$ sitting in degree $i$, respectively. The comparison isomorphism from $p$-adic Hodge theory gives $$ B_{\mathrm{HT}}\otimes V \cong B_{\mathrm{HT}}\otimes W. $$ Taking the degree $0$ part of this one obtains an isomorphism $$ \mathbb C_p\otimes V \cong \mathbb C_p\otimes W_0 \oplus \mathbb C_p(1-k)\otimes W_{k-1}, $$ which differs from Faltings' isomorphism above.

What is the precise relation between Faltings' Eichler-Shimura isomorphism and the general comparison isomorphism? Since I am not familiar with the techniques used by Faltings, I was not able to prove anything about this by myself.