how to design a coaxial cavity with specific reactance slope

These are ordinary quarter wave transmission lines. They are sometimes made slightly shorter so that the capacitor between the open end and ground can change the resonant frequency. The reactance slope is determined by the loaded Q. You control this with the source and load impedance.

the reactance slope is essential in filter design especially in bandstop
filter design.
The reactance slope is not determined by Q factor.
In fact,the L and C determine reactance slope.for example, if L is larger and C is smaller,then the reactance slope is larger.
reactace X=wL-1/wc, w0=sqrt(1/L/C),
reactance slope=dX/dw

so to exactly design L and C is the way to design w0 and reactance slope.
but in capacitor loaded coaxial cavity,
the X=Z0tan(beta*d)-1/wc
beta=2*pi/lamda,
but the L is not simply given. the X(w) function is somewhat complex.