Is there a non-deterministic linear time algorithm for CNF-SAT? The decision problem CNF-SAT can be described as follows:

Input: A boolean formula $\phi$ in conjunctive normal form.

Question: Does there exist a variable assignment that satisfies $\phi$?

I'm considering several different approaches for solving CNF-SAT with a non-deterministic two-tape Turing machine.

I believe that there is an NTM that solves CNF-SAT in $n \cdot \texttt{poly}(\log(n))$ steps.

Question: Is there an NTM that solves CNF-SAT in $O(n)$ steps?

Any relevant references are appreciated even if they only provide near linear time non-deterministic approaches.
This is only an extended comment.
A few times ago I asked (myself :-) how fast a multitape NTM that accepts a (reasonably encoded) NP-complete language can be. I came up with this idea: 

3-SAT remains NP-complete even if variables are represented in unary. In particular we can convert a clause - suppose $(x_i \lor \neg x_j \lor x_k)$ - of an arbitrary 3-SAT formula $\varphi$ on $n$ variables and $m$ clauses in a sequence of characters over alphabet $\Sigma = \{ +, -, 1 \}$ in which every variable occurrence is represented in unary:

$ + 1^{i} 0,- 1^{j} ,+ 1^{k} $

For example, $(x_2 \lor -x3 \lor +4)$ can be converted to:

+110-1110+11110


So we can convert a 3-SAT formula $\varphi_i$ in an equivalent string $U(\varphi_i)$ concatenating its clauses. The language $L_U = \{ U(\varphi_i) \mid \varphi_i \in 3-SAT \}$ is NP-complete.

A 2-tape NTM can decide if a string $x \in L_U$ in time $2|x|$ in this way.


the first head scans the input from left to right and with the internal logic it keeps track when it enters or exit a clause or reach the end of the formula. Whenever it finds a $+$ or $-$, the second head starts moving right with it on the $1^i$ that represents $x_i$. At the end of $1^i$, if the second head is on a $0$ then it guesses a truth value $+$ or $-$ (it makes an assignment) and writes it on the second tape; if it finds a $+$ or $-$ then that variable has already been assigned a value;
in both cases, using the internal logic, the NTM matches the truth value under the second head (the assignment) with the last seen $+$ or $-$; if they match then the clause is satisfied;
then the second head can return to the rightmost cell;
with the internal logic the NTM can keep track if all clauses are satisfied while the first head moves towards the end of the input.


Example:

Tape 1 (formula)    Tape 2 (variable assignments)
+110-1110+11110...  0000000000000...
^                   ^
+110-1110+11110...  0000000000000...
 ^                  ^
+110-1110+11110...  0000000000000...
  ^                  ^
+110-1110+11110...  0+00000000000... first guess set x2=T; matches +
  ^                  ^               so remember that current clause is satisfied
+110-1110+11110...  0+00000000000... 
  ^                  ^
...
+110-1110+11110...  0+00000000000... 
    ^               ^
...
+110-1110+11110...  0++0000000000... second guess set x3=T
       ^              ^              don't reject because current
                                     clause is satisfied (and in every
                                     case another literal must be parsed)


The time can be reduced to $|x|$ if we add some redundant symbols to the clause representation:

$ + 1^{i} 0^i,- 1^{j} 0^j ,+ 1^{k} 0^k \; ... \; \text{+++}$

($\text{+++}$ marks the end of the formula)

In this way the second head can return to the leftmost cell while the first scans the $0^i$ part.
Using $\text{++}$ as a clause delimiter and $\text{+++}$ as a marker for the end of the formula we can use same representation for CNF formulas with an arbitrary number of literals per clause.