Try using a Pickle file to store the Gekko model. If the Gekko model archive exists then it is read back into Python.
from os.path import exists
import pickle
import numpy as np
from gekko import GEKKO
import matplotlib.pyplot as plt
if exists('m.pkl'):
# load model from subsequent call
m = pickle.load(open('m.pkl','rb'))
m.solve()
else:
# define model the first time
m = GEKKO()
m.time = np.linspace(0,20,41)
m.p = m.MV(value=0, lb=0, ub=1)
m.v = m.CV(value=0)
m.Equation(5*m.v.dt() == -m.v + 10*m.p)
m.options.IMODE = 6
m.p.STATUS = 1; m.p.DCOST = 1e-3
m.v.STATUS = 1; m.v.SP = 40; m.v.TAU = 5
m.options.CV_TYPE = 2
m.solve()
pickle.dump(m,open('m.pkl','wb'))
plt.figure()
plt.subplot(2,1,1)
plt.plot(m.time,m.p.value,'b-',lw=2)
plt.ylabel('gas')
plt.subplot(2,1,2)
plt.plot(m.time,m.v.value,'r--',lw=2)
plt.ylabel('velocity')
plt.xlabel('time')
plt.show()
Each cycle of the controller, the plot updates with the automatic time-shift of the initial condition.

This is similar to what happens in a loop with a combined MHE and MPC. As long as you include everything in the Pickle file, it should reload on the next cycle.

Here is the example code for MHE and MPC.