Everything will sag. The question is if it will sag more than the Earth will curve.

Assuming the linear density is constant, it will follow an equation of the form y=a*cosh(x/a), or if shifted to have the centre at (0,0), will be y=a*cosh(x/a) - a.

This means the end will be at a height of a*cosh(l/a) - a above the centre (where l is the distance from the edge to the centre).

In this equation, a is equal to T/(lambda*g), i.e. the tension divided by the linear density and gravitational acceleration.

And if we divide the top and bottom by the area, this gives us the stress and density instead of tension and linear density.

And if we multiple a by g, we end up with stress on density, which is the specific strength.

If you take a string 2 km long, the distance from the edge to the centre will be 1 km.

The drop over this distance (equal to the bulge at the centre of 2 km) will be ~78.5 mm.

To have the string only sag to the 78.5 mm, you need a to be equal to ~6371000. This corresponds to 62.4 MPa m^3/kg or 62.4 GPa cm^3/g or 62400 kN m/kg.

This site has some nice lists of specific strengths:

https://www.engineeringtoolbox.com/engineering-materials-properties-d_1225.htmlConvienently, the units they have for Specific strength are the same as above.

But the highest value they have is 1.08, just roughly 1.7% of what is needed.

Wikipedia has a bigger list:

https://en.wikipedia.org/wiki/Specific_strengthBut it uses different units. In the units they use we need ~62400. That is only met by graphene, at which point you need to consider g. With g=9.8, graphene just beats the requirement and we get the sag 0.02 mm less than the bulge. With g=9.81, we instead get 0.06 mm more sag than the bulge.

And that is with the graphene at its limits. So if you go slightly tighter, you snap it.

Now, vibranium allegedly has a specific strength of 5835 kN m/kg, making it better than carbon fibre, but worse than graphene.