Let $E\rightarrow B$ be a bundle with inner product. Let $D(E)$ and $S(E)$ denote the disk bundle and the sphere bundle, respectively.
Then $D(E)\rightarrow B$ and $S(E)\rightarrow B$ are fiber bundles with fibers $D^n$ and $S^{n-1}$. Since $S(E)$ is weakly homotopy equivalent to $E-0$ and analogously for $D(E)$ we can concider following diagram:
where the horizontal arrows are inclusions. Now we can concider mapping cones on the both rows
$$ E-0\xrightarrow{j_1} E \xrightarrow{j_2} C(j_1)\xrightarrow{j_3} C(j_2)\rightarrow\ldots$$ $$ S(E)\xrightarrow{i_1} D(E) \xrightarrow{i_2} C(i_1)\xrightarrow{i_3} C(i_2)\rightarrow\ldots$$
Here is my question, can we conclude from above diagram that the mapping cones have weak equivalence, namely, that we have $C(i_k)\xrightarrow{\simeq} C(j_k)$ for all $k$?