(My solution has a trivial initial observation, an easy Case 1 and Case 2 divided in two sub-cases)

We analyze the possible outcomes from this configuration but can already discard

all configurations in which the lower "2"-cell has vertical lines, or horizontal lines,

in both edges.

Also, this configuration

.-------------------
| .   .   .   .   . 
|           2       
| .   .   .   .   . 
|       2           
| .___.   .   .   . 
| | 2               
|A.   .   .   .   . 
| x                 
| .   .   .   .   . 

and this

.-------------------
| .   .   .   .   . 
|           2       
| .   .   .   .   . 
|       2           
| .   .   .   .   . 
|   2 |             
|A.___.   .   .   . 
| x                 
| .   .   .   .   . 

are clearly impossible,

(dead end at corner A).

Case 1:

.-------------------
| .   .   .   .   . 
|           2       
| .   .   .   .   . 
|       2           
| .___.   .   .   . 
|   2 |             
| .   .   .   .   . 
| x                 
| .   .   .   .   . 
    

Now the following is forced

.-------------------
| .   .___.___.   . 
|         x 2 |     
| .   .___. x .   . 
| |   x 2 |         
| .___. x .   .   . 
|   2 |             
| .   .   .   .   . 
| x                 
| .   .   .   .   . 
    

and this leads to an impossibility in the cell at the corner, reached by three lines.

.-------------------
| .   .   .   .   . 
|           2       
| .   .   .   .   . 
|       2           
| .   .   .   .   . 
| | 2               
| .___.   .   .   . 
| x                 
| .   .   .   .   . 

The following is a forced move

.-------------------
| .   .   .   .   . 
|           2       
| .   .B  .   .   . 
| |     2           
| . x .A  .   .   . 
| | 2 x             
| .___.   .   .   . 
| x                 
| .   .   .   .   . 

Case 2.1: If a vertical line exists at AB, it forces

.-------------------
| .   .   .   .   . 
|     |     2       
| .C  .   .   .   . 
| |   | 2           
| . x .___.D  .   . 
| | 2 x             
| .___.   .   .   . 
| x                 
| .   .   .   .   . 

and any continuation for the line meeting corner C will create a loop (impossible).

Case 2.2: the other possibility is to have the edge AB blocked, and then the bottom edge AD

of the "2"-cell in the middle needs to be blocked too, forcing the following configuration to be reached.

.-------------------
| .   .___.___.   . 
|         x 2 |     
| .   .___. x .   . 
| |   x 2 |         
| . x . x .   .   . 
| | 2 x             
| .___.   .   .   . 
| x                 
| .   .   .   .   . 

Now, we have an impossible situation in the cell at the corner, reached by three lines. □