Question: Unable to accurately solve an ODE BVP with delay terms?

Delay differential equations in Chebfun lists 15 examples "taken from the literature". Many of them can be (numerically) solved in Maple without difficulty, yet when I attempt to solve the Example 8 in the above link, Maple's internal solver `dsolve/numeric` just halts with an error. 

plots:-odeplot(dsolve({D(u)(t) + u(t)**2 + 2*u(1/2*t) = 1/2*exp(t), u(0) = u(1/3)}, type = numeric, range = 0 .. 1/3), size = ["default", "golden"]);
Error, (in dsolve/numeric) delay equations are not supported for bvp solvers

Even if I guess an initial (or final) value artificially, the solution is still less reliable (For instance, what is the approximate endpoint value? 0.26344 or 0.2668?): 

restart;
dde := D(u)(t) + u(t)**2 + u(t/2)*2 = exp(t)/2:
x__0 := 2668/10000:
sol0 := dsolve([dde, u(0) = x__0], type = numeric, 'delaymax' = 1/6, range = 0 .. 1/3):
plots['odeplot'](sol0, [[t, u(t)], [t, x__0]], 'size' = ["default", "golden"]);

x__1 := 26344/100000:
sol1 := dsolve([dde, u(1/3) = x__1], type = numeric, 'delaymax' = 1/6, range = 0 .. 1/3):
plots['odeplot'](sol1, [[t, u(t)], [t, x__1]], size = ["default", "golden"]);

Compare:  (Note that the reference numerical solution implies that its minimum should be no less than 0.258 (Is this incorrect?).).

And actually, the only known constraint is simply u(0)＝u(⅓) (so neither value is known beforehand). Can Maple process this boundary condition automatically (that is, without the need for manual preprocessing and in absence of any other prior information)?
I have read the help page How to | Numeric Delay Differential Equations and Numerical Solution of Difficult ODE Boundary Value Problems, but it appears that those techniques are more or less ineffective here. So, how do I solve such a "first order nonlinear 'BVP' with pantograph delay" in Maple?