Define a reproducing kernel on the Euclidean ball in $\mathbb{C}^d$ by $$k(z,w)=\frac{1}{1-\langle z,w\rangle}+\frac{1}{1-\langle w,z\rangle}-1.$$ Call the corresponding real reproducing kernel Hilbert space the harmonic Drury-Arveson space $\mathcal{HDA}_d.$
We desire a concrete pair of $h_1, h_2 \in \mathcal{HDA}_d$ positive functions such that $\langle h_1,h_2\rangle <0$ for $d\geq 2.$
I will try to prove that such functions do not exist. Suppose that $f,g$ are positive (pluri)harmonic functions in the pluri harmonic Drury Arveson space $\mathcal{H}DA_d$ such that $ \langle f ,g \rangle < 0 $. By taking a small dilation of $f$ and $g$ we can assume that they are pluri harmonic in a ball of radius $1+\varepsilon$ for some $\varepsilon >0$ and they do not vanish on the closed unit ball $\mathbb{B}_d$.
Define the function $\mathcal{R}^{d-1}f $ through the equation; $$ f(z) = \int_{\partial \mathbb{B}_d} \frac{1-|\langle z, \zeta\rangle | ^2}{|1-\langle z, \zeta \rangle|^2 } \mathcal{R}^{d-1}f (\zeta) d\sigma(\zeta)=:\int_{\partial \mathbb{B}_d} C(z,\zeta) \mathcal{R}^{d-1} f (\zeta) d\sigma(\zeta) \,\, \forall z \in \mathbb{B}_d $$ (Just expand everythin in series and equate the coefficients, this way you define the coefficients of $\mathcal{R}^{d-1}f$) . In particular this formula implies that $ \mathcal{R}^{d-1} f \geq 0 $. That is because if we assume that there exists some point $\zeta_0 \in \partial \mathbb{B}_d$ such that $ \mathcal{R}^{d-1}f(\zeta_0) < 0 $. Then by continuity of $\mathcal{R}^{d-1} f$ there exists a Koranyi ball $B(\zeta_0, r_0) = \{ \zeta \in \partial \mathbb{B}_d : |1-\langle \zeta , \zeta_0 \rangle |^\frac12 < r_0 \} $ on which $\mathcal{R}^{d-1} f$ is strictly negative. This leads to a contradiction because the kernel $C(z,\zeta_0)$ as $\zeta_0 $ approaches radialy $\zeta \in \partial \mathbb{B}_d $ converges uniformly to zero outside any Koranyi ball $ B(\zeta_0,r_0)$. Finally be maximum principle $\mathcal{R}^{d-1} f (z) \geq 0, \,\, \forall z \in \overline{\mathbb{B}_d}$. Expanding again in series of homogeneous pluriharmonic polynomials it can be verified that $$ \langle f, g \rangle = \langle \mathcal{R}^{d-1} f , g \rangle_{L^2(\partial \mathbb{B}_d, d\sigma)} \geq 0. $$
