Conjectured sharp bounds for the expected linear complexity of multisequences

Let $q$ be the alphabet size, let $n$ be the length, and let $m$ be the number of component sequences. Write $E_n^{(m)}$ for the expected linear complexity of an $m$-multisequence of length $n$. Conjectured bounds. In view of numerical results, one has

$$\left\lceil\frac{nm}{m+1}\right\rceil-1+O(1/n) \le E_n^{(m)} \le \left\lceil\frac{nm}{m+1}\right\rceil+1+O(1/n).$$

The preceding theorem establishes the central estimate $E_n^{(m)}=\left\lceil\frac{mn}{m+1}\right\rceil+O(1)$; the conjecture proposes a substantially sharper description of the expected value, supported by numerical evidence.