Is $L \subset 1NL$ when $L \neq NL$? A log-space Turing machine has a read-only input tape, a write-only output tape and uses at most $O(\log n)$ space in its read-write work tapes. The classes $L$ and $NL$ contain those languages which are decided by deterministic or nondeterministic log-space Turing machines, respectively. The two-way Turing machines may move their head on the input tape into two-way (left and right directions) while the one-way Turing machines are not allowed to move the input head on the input tape to the left. 

Hartmanis and Mahaney have investigated the classes $1L$ and $1NL$ of languages recognizable by deterministic one-way log-space Turing machine and nondeterministic one-way log-space Turing machine, respectively. They have shown that $1NL \subseteq L$ if and only if $L=NL$.

See the paper here(it is free to download):

https://ecommons.cornell.edu/handle/1813/6253

I wonder this question:

Is $L \subset 1NL$ when $L \neq NL$?

Moreover, I wonder this another question:

Is there any reference that shows whether at least one of the options $L \subset 1NL$ or $1NL \subset L$ or $L = 1NL$ should be true?

Thanks in advance!!!
Define the language $BACKPOINTER$ to have words of length $n+t\log n$, divided to $1+t$ parts, one of length $n$ and the rest of length $\log n$, by commas such that $BACKPOINTER=\{(x,p_1,\ldots,p_t)\mid x_{p_i}=1 \forall i\}$.
It should follow from some standard one-way communication complexity bound that $BACKPOINTER$ needs at least $t$ bits of memory stored after $x$ is read. So for any $t\gg \log n$, $BACKPOINTER\in L\setminus 1NL$.

ps. Note that $BACKPOINTER\in 1co\text{-}NL$.