Let $G=\langle x, y, z| xyx^{-1}=zy, xzx^{-1}=z, yz=zy\rangle$, denote $l^1(G)^{\times}$ to be the set of units in $l^1(G)$, which we have considered as a ring with multiplication defined by the usual convolution, i.e., $(\sum_{g\in G}\lambda_gg)(\sum_{h\in G}\mu_hh)=\sum_{g, h\in G}\lambda_g\mu_hgh$.

Can we find $l=p_1(y, z)x^{n_1}+\cdots p_k(y,z)x^{n_k}\in l^1(G)^{\times}$ such that $\sum_{i=1}^k2^{n_i}p_i(y,z)(1-z^{n_i}y)=0$?

Here, $\forall~ 1\leq i\leq k, ~p_i(y,z)\in \mathbb{Z}G$ and $n_1<\cdots<n_k\in\mathbb{Z}$ to be determined. Note that the group element $x$ does not appear in $p_i(y, z)$.

Remarks:

1, This problem was asked previously in MSE, but no answer appeared, so I think it might be suitable for MO.

2, This problem is related to a modified Ore condition. I want to show that $l$ does not exist, suppose it exists, then I have considered the natural quotient map $\phi: G\to H=G/\langle z^2\rangle$. Note that it would induce a map $\phi: l^1(G)^{\times}\to l^1(H)^{\times}$, then $\phi(l)\in l^1(H)^{\times}$, but I still could not handle this..

Let $$G=\langle x, y, z\mid xyx^{-1}=zy, xzx^{-1}=z, yz=zy\rangle,$$ denote $l^1(G)^{\times}$ to be the set of units in $l^1(G)$, which we have considered as a ring with multiplication defined by the usual convolution, i.e., $(\sum_{g\in G}\lambda_gg)(\sum_{h\in G}\mu_hh)=\sum_{g, h\in G}\lambda_g\mu_hgh$.

Can we find $l=p_1(y, z)x^{n_1}+\cdots+ p_k(y,z)x^{n_k}\in l^1(G)^{\times}$ such that $$\sum_{i=1}^k2^{n_i}p_i(y,z)(1-z^{n_i}y)=0\;?$$

Here, $\forall~ 1\leq i\leq k, ~p_i(y,z)\in \mathbb{Z}G$ and $n_1<\cdots<n_k\in\mathbb{Z}$ to be determined. Note that the group element $x$ does not appear in $p_i(y, z)$.

Remarks:

1, This problem was asked previously in MSE, but no answer appeared, so I think it might be suitable for MO.

2, This problem is related to a modified Ore condition. I want to show that $l$ does not exist, suppose it exists, then I have considered the natural quotient map $\phi: G\to H=G/\langle z^2\rangle$. Note that it would induce a map $\phi: l^1(G)^{\times}\to l^1(H)^{\times}$, then $\phi(l)\in l^1(H)^{\times}$, but I still could not handle this..

Let $G=\langle x, y, z| xyx^{-1}=zy, xzx^{-1}=z, yz=zy\rangle$, denote $l^1(G)^{\times}$ to be the set of units in $l^1(G)$, which we have considered as a ring with multiplication defined by the usual convolution, i.e., $(\sum_{g\in G}\lambda_gg)(\sum_{h\in G}\mu_hh)=\sum_{g, h\in G}\lambda_g\mu_hgh$.

Can we find $l=p_1(y, z)x^{n_1}+\cdots p_k(y,z)x^{n_k}\in l^1(G)^{\times}$ such that $\sum_{i=1}^k2^{n_i}p_i(y,z)(1-z^{n_i}y)=0$?

Here, $\forall~ 1\leq i\leq k, ~p_i(y,z)\in \mathbb{Z}G$ and $n_1< \cdots <n_k\in \mathbb{Z}$ to be determined. Note that the group element $x$ does not appear in $p_i(y, z)$.

Remarks:

1, This problem was asked previously in MSE, but no answer appeared, so I think it might be suitable for MO.

2, This problem is related to a modified Ore condition. I want to show that $l$ does not exist, suppose it exists, then I have considered the natural quotient map $\phi: G\to H=G/<z^2>$. Note that it would induce a map $\phi: l^1(G)^{\times}\to l^1(H)^{\times}$, then $\phi(l)\in l^1(H)^{\times}$, but I still could not handle this..

Let $G=\langle x, y, z| xyx^{-1}=zy, xzx^{-1}=z, yz=zy\rangle$, denote $l^1(G)^{\times}$ to be the set of units in $l^1(G)$, which we have considered as a ring with multiplication defined by the usual convolution, i.e., $(\sum_{g\in G}\lambda_gg)(\sum_{h\in G}\mu_hh)=\sum_{g, h\in G}\lambda_g\mu_hgh$.

Can we find $l=p_1(y, z)x^{n_1}+\cdots p_k(y,z)x^{n_k}\in l^1(G)^{\times}$ such that $\sum_{i=1}^k2^{n_i}p_i(y,z)(1-z^{n_i}y)=0$?

Here, $\forall~ 1\leq i\leq k, ~p_i(y,z)\in \mathbb{Z}G$ and $n_1<\cdots<n_k\in\mathbb{Z}$ to be determined. Note that the group element $x$ does not appear in $p_i(y, z)$.

Remarks:

1, This problem was asked previously in MSE, but no answer appeared, so I think it might be suitable for MO.

2, This problem is related to a modified Ore condition. I want to show that $l$ does not exist, suppose it exists, then I have considered the natural quotient map $\phi: G\to H=G/\langle z^2\rangle$. Note that it would induce a map $\phi: l^1(G)^{\times}\to l^1(H)^{\times}$, then $\phi(l)\in l^1(H)^{\times}$, but I still could not handle this..