60s stair case

[DR007]

Ive noticed the top stair tread in my 60s house is only supported by what seems two small wood block at either end. The top tread also does not land on any joist and would explain the creaking. Im due to lay a new upstairs floor and wondering is it ok to replace the top tread so that it stretches to the joist. Whilst doing this i would also fill the gap between the joist and riser with timber to further strengthen. The question is, does this sound sensible or will the entire flights of stairs be left unsupported whilst i carry this out? Thanks in advance

 

[blup]

Stiffen the joist by fixing a second joist beneath it running the full width and screwed/nailed into each end. Placing it upright will add stiffness. No need to infill the other gap imo.

 

[johnny2007]

Go ahead.
The strings keep the staircase there.

 

[DR007]

Go ahead.
The strings keep the staircase there.

Thanks mate, I have noticed the plaster around the strings cracking top to bottom which ive tried to emphasis in the last photo.
This could be due to general bashing, walls chases and new flooring in each room causing movement.
The previous owner has taken the easy route of badly filling with caulk.
I wasn't sure weather the strings would be secured to each side of the wall with fixings or if the entire flight of stairs hangs off the those two small blocks on each side as I've seen on some videos online.
I think I will pull it out with the help of a multi tool and find a thicker 18mm nosing to match the height of the new 18mm flooring.

 

[DR007]

Stiffen the joist by fixing a second joist beneath it running the full width and screwed/nailed into each end. Placing it upright will add stiffness. No need to infill the other gap imo.

The issue is the space is so narrow about 25mm. This means a) it would be akward to get a drill in at an angle to secure a piece of timber directly under the unsupported nosing and b) I feel it would require something thicker than 25mm.