The effect of dilution on the rate of a system in equilibrium
The rate forwards and back are the same at equilibrium  and differ before this is attained. Let the reaction be $A+B=C$. If each concentration is suddenly halved then the reaction is not necessarily any longer at equilibrium and in this case the rates forwards and back differ until a new equilibrium is reached. The figure shows this. The red curve is the forwards rate $k_1[A][B]$ and the green one $k_{-1}[C]$ the reverse. As you see the new equilibrium takes a short while to re-establish. (You can ignore the scales they depend on the particular rate constants and concentrations used but the behaviour is general.)
The figure you show is not fully supported by the calculation. The red curve (forwards rate)  agrees but the blue one does not.  If you do not halve the product concentration you get the  blue curve in the figure.